updated manuscript
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@ -63,6 +63,8 @@
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\newcommand{\Ne}{N}
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\newcommand{\Nb}{N_{\Bas}}
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\newcommand{\Ng}{N_\text{grid}}
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\newcommand{\nocca}{n_{\text{occ}^{\alpha}}}
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\newcommand{\noccb}{n_{\text{occ}^{\beta}}}
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\Ec}{E_\text{c}}
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@ -331,6 +333,7 @@ In the final step, we employ $\rsmu{}{}(\br{})$ within short-range density funct
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\end{equation}
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where $\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})$ is the two-body density associated with $\wf{}{\Bas}$
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\begin{equation}
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\label{eq:n2basis}
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\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})
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= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2},
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\end{equation}
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@ -352,7 +355,14 @@ which therefore can be rewritten as
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\iint r_{12}^{-1} \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2} = \iint \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{equation}
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intuitively motivating $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ as a potential candidate for an effective interaction.
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As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries and is necessarily \textit{finite} at $r_{12} = 0$ for an incomplete basis set.
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As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries.
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An important quantity to define is $\W{\wf{}{\Bas}}{}(\br{})$ which is the value of the effective interaction at $\br{}$ for opposite spins at coalescence
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\begin{equation}
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\label{eq:wcoal}
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\W{\wf{}{\Bas}}{}(\br{}) = \W{\wf{}{\Bas}}{}(\bx{},\Bar{\bx{}})
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\end{equation}
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and which is necessarily \textit{finite} at $r_{12} = 0$ for an \textit{incomplete} basis set.
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Of course, there exists \textit{a priori} an infinite set of functions satisfying \eqref{eq:int_eq_wee}, but thanks to its very definition one can show (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that
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\begin{equation}
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\label{eq:lim_W}
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@ -363,19 +373,22 @@ which therefore guarantees a physically satisfying limit. An important point her
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%=================================================================
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%\subsection{Range-separation function}
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%=================================================================
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\manu{Working on that paragraph}
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As we can map the Coulomb operator within a basis set $\Bas$ with a non divergent two-electron interaction, we can link the present theory with the RS-DFT which uses a smooth bounded two-electron interactions.
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In pracice, to be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$, we use the functionals developed in the field of RS-DFT thanks to we associate the effective interaction to a long-range interaction characterized by a range-separation function $\rsmu{}{}(\br{})$.
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Although this choice is not unique, the long-range interaction we have chosen is
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\begin{equation}
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\w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{}{}(\br{2}) r_{12}]}{ r_{12}} }.
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\end{equation}
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Ensuring that $\w{}{\lr,\rsmu{}{}}(\br{1},\br{2})$ and $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ have the same value at coalescence of opposite-spin electron pairs yields
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As we can map the Coulomb operator within a basis set $\Bas$ with a non divergent two-electron interaction, we can link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators.
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To do so, we choose a range-separation \textit{function} $\rsmu{\wf{}{\Bas}}{}(\br{})$
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\begin{equation}
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\label{eq:mu_of_r}
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\rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\bx{},\Bar{\bx{}}),
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\rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{}),
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\end{equation}
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where $(\bx{},\Bar{\bx{}})$ represents a couple of opposite-spin electrons at the same position $\br{}$.
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such that the long-range interaction $\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2})$
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\begin{equation}
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\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\wf{}{\Bas}}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\wf{}{\Bas}}{}(\br{2}) r_{12}]}{ r_{12}} }
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\end{equation}
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coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all points in ${\rm I\!R}^3$
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\begin{equation}
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\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{},\br{}) = \W{\wf{}{\Bas}}{}(\br{})\quad \forall \,\, \br{}.
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\end{equation}
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%More precisely, if we define the value of the interaction at coalescence as
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%\begin{equation}
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% \label{eq:def_wcoal}
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@ -422,10 +435,10 @@ where $(\bx{},\Bar{\bx{}})$ represents a couple of opposite-spin electrons at th
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%=================================================================
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%\label{sec:ecmd}
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we propose here to approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
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Once defined the range-separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$, we can use the functionals defined in the field of RS-DFT to approximate $\bE{}{\Bas}[\n{}{}]$. As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
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\begin{multline}
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\label{eq:ec_md_mu}
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\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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\\
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- \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
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\end{multline}
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@ -458,10 +471,10 @@ This makes them particularly well adapted to the present context where one aims
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%--------------------------------------------
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%\subsubsection{Local density approximation}
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%--------------------------------------------
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{}{}(\br{})$. Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ as
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\begin{equation}
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\label{eq:def_lda_tot}
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\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] \n{}{}(\br{}) \dbr{},
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\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\UEG}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
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\end{equation}
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where $\be{\UEG}{\sr}(\n{}{},\rsmu{}{})$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
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%In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
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@ -485,7 +498,7 @@ This represents a major computational saving without loss of performance as we e
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Therefore, the PBE complementary functional reads
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\begin{equation}
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\label{eq:def_pbe_tot}
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\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})] \n{}{}(\br{}) \dbr{}.
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\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}\big(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{}.
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\end{equation}
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Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\wf{}{\Bas}}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ where $\rsmu{\wf{}{\Bas}}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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@ -502,8 +515,13 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
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% \lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
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%\end{equation}
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%for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
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The computational bottleneck of the present approach is the computation of $\f{\wf{}{\Bas}}{}(\br{})$ [See Eqs.~\eqref{eq:fbasis} and \eqref{eq:mu_of_r}] at each quadrature grid point.
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By performing successive matrix multiplications, this step can be formally performed at $\order{\Ng \Nb^4}$ computational cost and $\order{\Ng \Nb^2}$ storage (where $\Ng$ is the number of points of the quadrature grid and $\Nb$ is the number of basis functions in $\Bas$), although sparsity and/or atomic-orbital-based algorithms could further reduce this cost.
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The computational bottleneck of the present approach is the computation of $\W{\wf{}{\Bas}}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. All through this paper, we use two-body density matrix of a single Slater determinant (typically HF) for $\Gam{rs}{tu}[\wf{}{\Bas}]$ and therefore the computational bottleneck reduces to the evaluation at each quadrature grid point of
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\begin{equation}
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\label{eq:fcoal}
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f_{\text{HF}}^{\Bas}(\br{}) = \sum_{p,q\in\Bas} \sum_{i\in \nocca} \sum_{j\in \noccb} \V{pq}{ij} \SO{p}{1} \SO{q}{2} \SO{i}{1} \SO{j}{2}
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\end{equation}
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which scales as $\Nb^2\times N_{elec}^2 \times \Ng$ and is embarassingly parallel. Within the present formulation, the bottleneck is the four-index transformation to obtaine the two-electron integrals on the MO basis which appear in \eqref{eq:fcoal}, but this step has in general to be performed before a correlated WFT calculations.
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When the four-index transformation become prohibitive, by performing successive matrix multiplications, one could rewrite the equations directly in the AO basis where it scales formally as $\order{\Ng \Nb^4}$ but where one can take advantage of the sparsity atomic-orbital-based algorithms to reach a linear scaling method.
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%=================================================================
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@ -513,16 +531,7 @@ As most WFT calculations are performed within the frozen-core (FC) approximation
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We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively.% and $\Cor \bigcap \Val = \O$.
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%According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$.
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Accounting solely for the valence electrons, Eq.~\eqref{eq:expectweeb} becomes
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\begin{equation}
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\label{eq:expectweebval}
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\mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{equation}
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where $\hWee{\Val}$, the valence part of the Coulomb operator, has a similar expression as $\hWee{\Bas}$ in Eq.~\eqref{eq:WeeB}.
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%\begin{equation}
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% \hWee{\Val} = \frac{1}{2} \sum_{ijkl \in \Val} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
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%\end{equation}
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Following the spirit of Eq.~\eqref{eq:fbasis}, we have
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Accounting solely for the valence electrons, we define
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\begin{multline}
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\label{eq:fbasisval}
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\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})
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@ -596,7 +605,7 @@ Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valenc
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\ce{C2} & exFCI\fnm[1] & 132.0 (-13.7 ) & 140.3 (-5.4 ) & 143.6 (-2.1 ) & 144.7 (-1.0 ) & 145.7 \\
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(cc-pVXZ) & exFCI+LDA\fnm[1] & 141.3 (-4.4 ) & 145.1 (-0.6 ) & 146.4 (+0.7 ) & 146.3 (+0.6 ) & \\
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& exFCI+PBE\fnm[1] & 145.7 (+0.0 ) & 145.7 (+0.0 ) & 146.3 (+0.6 ) & 146.2 (+0.5 ) & \\
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& CCSD(T)\fnm[1] & 129.2 (-16.2 ) & 139.1 (-6.3 ) & 143.0 (-2.4 ) & 144.2 (-1.2 ) & \\
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& CCSD(T)\fnm[1] & 129.2 (-16.2 ) & 139.1 (-6.3 ) & 143.0 (-2.4 ) & 144.2 (-1.2 ) & 145.4 \\
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& CCSD(T)+LDA\fnm[1] & 139.1 (-6.3 ) & 143.7 (-1.7 ) & 145.9 (+0.5 ) & 145.9 (+0.5 ) & \\
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& CCSD(T)+PBE\fnm[1] & 142.8 (-2.6 ) & 144.2 (-1.2 ) & 145.9 (+0.5 ) & 145.8 (+0.4 ) & \\ \\
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\ce{C2} & exFCI\fnm[2] & 131.0 (-16.1 ) & 141.5 (-5.6 ) & 145.1 (-2.0 ) & 146.1 (-1.0 ) & 147.1 \\
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@ -614,9 +623,9 @@ Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valenc
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\ce{O2} & exFCI\fnm[1] & 105.2 (-14.8 ) & 114.5 (-5.5 ) & 118.0 (-2.0 ) & 119.1 (-0.9 ) & 120.0 \\
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(cc-pVXZ) & exFCI+LDA\fnm[1] & 112.4 (-7.6 ) & 118.4 (-1.6 ) & 120.2 (+0.2 ) & 120.4 (+0.4 ) & \\
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& exFCI+PBE\fnm[1] & 117.2 (-2.8 ) & 119.4 (-0.6 ) & 120.3 (+0.3 ) & 120.4 (+0.4 ) & \\
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& CCSD(T)\fnm[1] & 103.9 (-16.1 ) & 113.6 (-6.4 ) & 117.1 (-2.9 ) & 118.6 (-1.4 ) & 120.0 \\
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& CCSD(T)+LDA\fnm[1] & 110.6 (-9.4 ) & 117.2 (-2.8 ) & 119.2 (-0.8 ) & 119.8 (-0.2 ) & \\
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& CCSD(T)+PBE\fnm[1] & 115.1 (-4.9 ) & 118.0 (-2.0 ) & 119.3 (-0.7 ) & 119.8 (-0.2 ) & \\ \\
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& CCSD(T)\fnm[1] & 103.9 (-16.1 ) & 113.6 (-6.0 ) & 117.1 (-2.5 ) & 118.6 (-1.0 ) & 119.6 \\
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& CCSD(T)+LDA\fnm[1] & 110.6 (-9.0 ) & 117.2 (-2.4 ) & 119.2 (-0.4 ) & 119.8 (+0.2 ) & \\
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& CCSD(T)+PBE\fnm[1] & 115.1 (-4.5 ) & 118.0 (-1.6 ) & 119.3 (-0.3 ) & 119.8 (+0.2 ) & \\ \\
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\ce{F2} & exFCI\fnm[1] & 26.7 (-12.3 ) & 35.1 (-3.9 ) & 37.1 (-1.9 ) & 38.0 (-1.0 ) & 39.0 \\
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(cc-pVXZ) & exFCI+LDA\fnm[1] & 30.4 (-8.6 ) & 37.2 (-1.8 ) & 38.4 (-0.6 ) & 38.9 (-0.1 ) & \\
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& exFCI+PBE\fnm[1] & 33.1 (-5.9 ) & 37.9 (-1.1 ) & 38.5 (-0.5 ) & 38.9 (-0.1 ) & \\
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@ -674,8 +683,9 @@ This molecular set has been exhausively studied in the last 20 years (see, for e
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%The reference values for the atomization energies are extracted from Ref.~\onlinecite{HauKlo-JCP-12} and corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
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As a method $\modX$ we employ either CCSD(T) or exFCI.
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Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
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We refer the interested reader to Refs.~\onlinecite{SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
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Throughout this study, we have $\modY = \HF$ as we use the Hartree-Fock (HF) one-electron density to compute the complementary energy.
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We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
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In the case of the CCSD(T) calculations, we have $\modY = \HF$ as we use the Restricted Open Shel Hartree-Fock (ROHF) one-electron density to compute the complementary energy, and for the CIPSI calculations we use the density of a converged variational wave function.
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For the definition of the interaction, we use a single Slater determinant in the ROHF basis.
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The CCSD(T) calculations have been performed with Gaussian09 with standard threshold values. \cite{g09}
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RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
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\titou{For the quadrature grid, we employ ... radial and angular points.}
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